Integrality relations for polygonal dissections

Aaron Abrams and James Pommersheim

Vol. 330 (2024), No. 2, 199–206

DOI: 10.2140/pjm.2024.330.199

Abstract

Given a parallelogram dissected into triangles, the area of any one of the triangles of the dissection is integral over the ring generated by the areas of the other triangles. Given a trapezoid dissected into triangles, the area of any triangle determined by either diagonal of the trapezoid is integral over the ring generated by the areas of the triangles in the dissection. In both cases, the integrality relations are invariant under deformation of the dissection.

The trapezoid theorem implies and provides a new context for Monsky’s equidissection theorem that a square cannot be dissected into an odd number of triangles of equal area. A corollary of these results is that the area polynomials for parallelograms we introduced and studied in previous work (2014; 2022; 2023) have all leading coefficients equal to $\pm 1$.

Keywords

integrality relation, dissection, Monsky polynomial

Mathematical Subject Classification

Primary: 52B45

Secondary: 51M25

Milestones

Received: 16 June 2023

Revised: 30 January 2024

Accepted: 11 May 2024

Published: 13 September 2024